`Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'`🔗

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Minimal Lean file

`law_pullCount_sumRewards_unique'`🔗

LemmaLearning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'

Details

No docstring.

theorem

Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {Ω : Type u_2} {Ω' : Type u_3} [DecidableEq 𝓐]
  {m𝓐 : MeasurableSpace 𝓐} {mΩ : MeasurableSpace Ω}
  {mΩ' : MeasurableSpace Ω'} {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P] {P' : MeasureTheory.Measure Ω'}
  [MeasureTheory.IsProbabilityMeasure P'] {alg : Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν] {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {A₂ : ℕ → Ω' → 𝓐} {R₂ : ℕ → Ω' → ℝ} {n : ℕ}
  [MeasurableSingletonClass 𝓐]
  (h1 : IsAlgEnvSeq A R alg (stationaryEnv ν) P)
  (h2 : IsAlgEnvSeq A₂ R₂ alg (stationaryEnv ν) P') :
  ProbabilityTheory.IdentDistrib
    (fun ω a => (pullCount A a n ω, sumRewards A R a n ω))
    (fun ω a => (pullCount A₂ a n ω, sumRewards A₂ R₂ a n ω)) P P'
Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {Ω : Type u_2}
  {Ω' : Type u_3} [DecidableEq 𝓐]
  {m𝓐 : MeasurableSpace 𝓐}
  {mΩ : MeasurableSpace Ω}
  {mΩ' : MeasurableSpace Ω'}
  {P : MeasureTheory.Measure Ω}
  [MeasureTheory.IsProbabilityMeasure P]
  {P' : MeasureTheory.Measure Ω'}
  [MeasureTheory.IsProbabilityMeasure P']
  {alg : Algorithm 𝓐 ℝ}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  [ProbabilityTheory.IsMarkovKernel ν]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → ℝ}
  {A₂ : ℕ → Ω' → 𝓐} {R₂ : ℕ → Ω' → ℝ}
  {n : ℕ} [MeasurableSingletonClass 𝓐]
  (h1 :
    IsAlgEnvSeq A R alg (stationaryEnv ν)
      P)
  (h2 :
    IsAlgEnvSeq A₂ R₂ alg
      (stationaryEnv ν) P') :
  ProbabilityTheory.IdentDistrib
    (fun ω a =>
      (pullCount A a n ω,
        sumRewards A R a n ω))
    (fun ω a =>
      (pullCount A₂ a n ω,
        sumRewards A₂ R₂ a n ω))
    P P'

Code

lemma _root_.Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique' [MeasurableSingletonClass 𝓐]
    (h1 : IsAlgEnvSeq A R alg (stationaryEnv ν) P)
    (h2 : IsAlgEnvSeq A₂ R₂ alg (stationaryEnv ν) P') :
    IdentDistrib (fun ω a ↦ (pullCount A a n ω, sumRewards A R a n ω))
      (fun ω a ↦ (pullCount A₂ a n ω, sumRewards A₂ R₂ a n ω)) P P'

Type uses (5)

Body uses (7)

Used by (2)

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Proof

by
  have hA := h1.measurable_action
  have hR := h1.measurable_feedback
  have hA2 := h2.measurable_action
  have hR2 := h2.measurable_feedback
  constructor
  · refine Measurable.aemeasurable ?_
    rw [measurable_pi_iff]
    exact fun a ↦ Measurable.prod (by fun_prop) (measurable_sumRewards hA hR _ _)
  · refine Measurable.aemeasurable ?_
    rw [measurable_pi_iff]
    exact fun a ↦ Measurable.prod (by fun_prop) (measurable_sumRewards hA2 hR2 _ _)
  have h_unique := isAlgEnvSeq_unique h1 h2
  let f := fun (p : ℕ → 𝓐 × ℝ ) (a : 𝓐) ↦ (∑ i ∈ range n, if (p i).1 = a then 1 else 0,
    ∑ i ∈ range n, if (p i).1 = a then (p i).2 else 0)
  have hf : Measurable f := by
    rw [measurable_pi_iff]
    intro a
    refine Measurable.prod ?_ ?_
    · simp only [f]
      refine measurable_sum _ fun i hi ↦ Measurable.ite ?_ (by fun_prop) (by fun_prop)
      exact (measurableSet_singleton _).preimage (by fun_prop)
    · simp only [f]
      refine measurable_sum _ fun i hi ↦ Measurable.ite ?_ (by fun_prop) (by fun_prop)
      exact (measurableSet_singleton _).preimage (by fun_prop)
  have h_eq_comp : (fun ω a ↦ (pullCount A a n ω, sumRewards A R a n ω))
      = f ∘ (trajectory A R) := by
    ext ω a : 2
    rw [pullCount_eq_comp (R := R), sumRewards_eq_comp]
    grind
  have h_eq_comp2 : (fun ω a ↦ (pullCount A₂ a n ω, sumRewards A₂ R₂ a n ω))
      = f ∘ (trajectory A₂ R₂) := by
    ext ω a : 2
    rw [pullCount_eq_comp (R := R₂), sumRewards_eq_comp]
    grind
  rw [h_eq_comp, h_eq_comp2, ← Measure.map_map hf, h_unique, Measure.map_map hf,
    ← h_eq_comp2]
  · rw [measurable_pi_iff]
    exact fun n ↦ Measurable.prodMk (hA2 n) (hR2 n)
  · rw [measurable_pi_iff]
    exact fun n ↦ Measurable.prodMk (hA n) (hR n)

Dependency graph

Type dependencies (5)

`Algorithm`🔗

StructureLearning.Algorithm

Details

A stochastic, sequential algorithm.

structure

Learning.Algorithm.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Algorithm (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : ℕ) → Kernel (Iic n → 𝓐 × 𝓨) 𝓐
  /-- The policy is a Markov kernel. -/
  [h_policy : ∀ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure 𝓐
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]

Used by (216)

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`IsAlgEnvSeq`🔗

StructureLearning.IsAlgEnvSeq

Details

An algorithm-environment sequence: a sequence of actions and feedbacks generated by an algorithm interacting with an environment.

structure

Learning.IsAlgEnvSeq.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨)
  (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure P] : Prop
Learning.IsAlgEnvSeq.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  {mΩ : MeasurableSpace Ω} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨)
  (env : Environment 𝓐 𝓨)
  (P : MeasureTheory.Measure Ω)
  [MeasureTheory.IsFiniteMeasure P] : Prop

Code

structure IsAlgEnvSeq
    (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (alg : Algorithm 𝓐 𝓨) (env : Environment 𝓐 𝓨)
    (P : Measure Ω) [IsFiniteMeasure P] : Prop where
  /-- The action sequence is measurable. -/
  measurable_action n : Measurable (A n) := by fun_prop
  /-- The feedback sequence is measurable. -/
  measurable_feedback n : Measurable (Y n) := by fun_prop
  /-- The first action has the correct law. -/
  hasLaw_action_zero : HasLaw (fun ω ↦ (A 0 ω)) alg.p0 P
  /-- The first feedback has the correct conditional distribution. -/
  hasCondDistrib_feedback_zero : HasCondDistrib (Y 0) (A 0) env.ν0 P
  /-- The next action has the correct conditional distribution given the history. -/
  hasCondDistrib_action n :
    HasCondDistrib (A (n + 1)) (history A Y n) (alg.policy n) P
  /-- The next feedback has the correct conditional distribution given the history and
  next action. -/
  hasCondDistrib_feedback n :
    HasCondDistrib (Y (n + 1)) (fun ω ↦ (history A Y n ω, A (n + 1) ω))
      (env.feedback n) P

Type uses (3)

Used by (111)

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`stationaryEnv`🔗

DefinitionLearning.stationaryEnv

Details

A stationary environment, in which the distribution of the next feedback depends only on the last action.

def

Learning.stationaryEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (ν : ProbabilityTheory.Kernel 𝓐 𝓨)
  [ProbabilityTheory.IsMarkovKernel ν] : Environment 𝓐 𝓨
Learning.stationaryEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (ν : ProbabilityTheory.Kernel 𝓐 𝓨)
  [ProbabilityTheory.IsMarkovKernel ν] :
  Environment 𝓐 𝓨

Code

def stationaryEnv (ν : Kernel 𝓐 𝓨) [IsMarkovKernel ν] : Environment 𝓐 𝓨 := obliviousEnv fun _ ↦ ν

Type uses (1)

Environment

Body uses (1)

obliviousEnv

Used by (81)

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`pullCount`🔗

DefinitionLearning.pullCount

Details

Number of times action a was chosen up to time t (excluding t).

def

Learning.pullCount.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ
Learning.pullCount.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (a : 𝓐)
  (t : ℕ) (ω : Ω) : ℕ

Code

noncomputable
def pullCount (A : ℕ → Ω → 𝓐) (a : 𝓐) (t : ℕ) (ω : Ω) : ℕ :=
  #(filter (fun s ↦ A s ω = a) (range t))

Used by (146)

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`sumRewards`🔗

DefinitionLearning.sumRewards

Details

Sum of rewards obtained when pulling action a up to time t (exclusive).

def

Learning.sumRewards.{u_1, u_3} {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐) (R' : ℕ → Ω → ℝ) (a : 𝓐) (t : ℕ)
  (ω : Ω) : ℝ
Learning.sumRewards.{u_1, u_3}
  {𝓐 : Type u_1} {Ω : Type u_3}
  [DecidableEq 𝓐] (A : ℕ → Ω → 𝓐)
  (R' : ℕ → Ω → ℝ) (a : 𝓐) (t : ℕ)
  (ω : Ω) : ℝ

Code

def sumRewards (A : ℕ → Ω → 𝓐) (R' : ℕ → Ω → ℝ) (a : 𝓐) (t : ℕ) (ω : Ω) : ℝ :=
  ∑ s ∈ range t, if A s ω = a then R' s ω else 0

Used by (44)

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All dependencies, transitively (3)

`Environment`🔗

StructureLearning.Environment

Details

A stochastic environment.

structure

Learning.Environment.{u_4, u_5} (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] : Type (max u_4 u_5)
Learning.Environment.{u_4, u_5}
  (𝓐 : Type u_4) (𝓨 : Type u_5)
  [MeasurableSpace 𝓐]
  [MeasurableSpace 𝓨] : Type (max u_4 u_5)

Code

structure Environment (𝓐 𝓨 : Type*) [MeasurableSpace 𝓐] [MeasurableSpace 𝓨] where
  /-- Distribution of the next observation as function of the past history. -/
  feedback : (n : ℕ) → Kernel ((Iic n → 𝓐 × 𝓨) × 𝓐) 𝓨
  /-- The feedback kernels are Markov kernels. -/
  [h_feedback : ∀ n, IsMarkovKernel (feedback n)]
  /-- Distribution of the first observation given the first action. -/
  ν0 : Kernel 𝓐 𝓨
  /-- The initial observation kernel is a Markov kernel. -/
  [hp0 : IsMarkovKernel ν0]

Used by (128)

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`history`🔗

DefinitionLearning.history

Details

History of the algorithm-environment sequence up to time n.

def

Learning.history.{u_1, u_2, u_3} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨
Learning.history.{u_1, u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} (A : ℕ → Ω → 𝓐)
  (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) :
  ↥(Finset.Iic n) → 𝓐 × 𝓨

Code

def history (A : ℕ → Ω → 𝓐) (Y : ℕ → Ω → 𝓨) (n : ℕ) (ω : Ω) : Iic n → 𝓐 × 𝓨 :=
  fun i ↦ (A i ω, Y i ω)

Used by (72)

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`obliviousEnv`🔗

DefinitionLearning.obliviousEnv

Details

An oblivious environment, in which the distribution of the next feedback depends only on the last action, but in a possibly time-dependent manner.

def

Learning.obliviousEnv.{u_1, u_2} {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (ν n)] : Environment 𝓐 𝓨
Learning.obliviousEnv.{u_1, u_2}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {m𝓨 : MeasurableSpace 𝓨}
  (ν : ℕ → ProbabilityTheory.Kernel 𝓐 𝓨)
  [∀ (n : ℕ),
      ProbabilityTheory.IsMarkovKernel
        (ν n)] :
  Environment 𝓐 𝓨

Code

def obliviousEnv (ν : ℕ → Kernel 𝓐 𝓨) [∀ n, IsMarkovKernel (ν n)] : Environment 𝓐 𝓨 where
  feedback n := (ν (n + 1)).prodMkLeft _
  ν0 := ν 0

Type uses (1)

Environment

Used by (10)

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Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'🔗

law_pullCount_sumRewards_unique'🔗

Algorithm🔗

IsAlgEnvSeq🔗

stationaryEnv🔗

pullCount🔗

sumRewards🔗

Environment🔗

history🔗

obliviousEnv🔗

`Learning.IsAlgEnvSeq.law_pullCount_sumRewards_unique'`🔗

`law_pullCount_sumRewards_unique'`🔗

`Algorithm`🔗

`IsAlgEnvSeq`🔗

`stationaryEnv`🔗

`pullCount`🔗

`sumRewards`🔗

`Environment`🔗

`history`🔗

`obliviousEnv`🔗